Certain very large cardinals are not created in small forcing extensions

Richard Laver
2007 Annals of Pure and Applied Logic  
The large cardinal axioms of the title assert, respectively, the existence of a nontrivial elementary embedding j : V λ → V λ , the existence of such a j which is moreover Σ 1 n , and the existence of such a j which extends to an elementary j : V λ+1 → V λ+1 . It is known that these axioms are preserved in passing from a ground model to a small forcing extension. In this paper the reverse directions of these preservations are proved. Also the following is shown (and used in the above proofs in
more » ... lace of using a standard fact): if V is a model of ZFC and V [G] is a P-generic forcing extension of V , then in V [G], V is definable using the parameter A property which has been verified for most large cardinal axioms is that the satisfaction of the axiom for a cardinal κ cannot be created or destroyed in a small forcing extension -if V is a model of ZFC in which κ is a cardinal and P is a partial ordering with = P < κ, LC(κ) is a large cardinal axiom about κ, and G ⊆ P is V -generic, then The proof of ( * ) when LC(κ) is "κ is a measurable cardinal" is due to Levy and Solovay [10] . See Jech [4], Hamkins and Woodin [3] and Hamkins [2] for instances of ( * ) for other large cardinal axioms. Most of the large cardinal axioms from measurable cardinals upwards assert the existence of elementary embeddings j from one transitive set or class to another, where the large cardinal κ is cr( j), the least ordinal moved by j. The proofs of the left to right direction of ( * ) for such axioms show that every j witnessing LC(κ) in V lifts to aĵ witnessing LC(κ) in V [G]. And the right to left directions, when LC(κ) is not too strong, are partial converses: if k witnesses LC(κ) in V [G], then for a canonical embedding k induced by k for LC(κ), k witnesses LC(κ) in V [G] and k V witnesses LC(κ) in V .
doi:10.1016/j.apal.2007.07.002 fatcat:coqlsq2xpnd2pdfkn56hjq7mji